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\begin{document}

\title{FDC prediction and inference: insights from the fusion of machine
learning methods and basin characteristic factors}



\author[1]{Yu Zhou}%
\author[1]{Wuyi Wan}%
\affil[1]{Zhejiang University}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\selectlanguage{english}
\begin{abstract}
This paper aims to solve the problem of accurately estimating flow
duration curves (FDC) in catchments lacking diachronic flow data. Based
on 645 sets of observed data in the middle and lower reaches of the
Yangtze River (YZR), which include 22 basin characteristic variables,
eight machine learning (ML) models (SVM, RF, BPNN, ELM, XGB, RBF,
PSO-BP, GWO-BP) were integrated to predict the FDC (quantiles of flow
rate corresponding to 15 exceedance probabilities were studied), after
which the model most suitable for predicting was determined. Finally,
the SHapley Additive exPlanation (SHAP) method was used to determine and
quantify the impact of various input variables on different quantiles
and the degree of that influence. Results indicate that: (1) The GWO-BP
model is the best ML model for predicting FDC among the eight, having
good prediction performances throughout the entire duration with
determination coefficients (R2) on the testing set of 0.86 to 0.94 and
Nash-Sutcliff criterion (NSE) of 0.78 to 0.94. (2) The ML model (BPNN)
optimized using swarm intelligence can effectively predict FDC. (3) The
predictive impact of variables on different quantiles varies, with and
BFI\_mean contributes significantly to predicting FDC. The former has a
negative effect on the prediction result and has better contribution to
predicting higher flow rate (i.e., having higher accuracy in predicting
the upper tail of FDC), whereas the latter is the opposite. SHAP's
explanations are consistent with the physical model, revealing local
interactions between predictive factors. The results demonstrate that
the method proposed in this paper can greatly improve the prediction
accuracy and is innovative and valuable in model interpretation and
factor selection.%
\end{abstract}%



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\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

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\end{figure}

\textbf{FDC prediction and inference: insights from the fusion of
machine learning methods and basin characteristic factors}

Yu Zhou \textsuperscript{1}, Wuyi Wan \textsuperscript{1}

\textsuperscript{1} Dept. of Hydraulic Engineering, College of Civil
Engineering and Architecture, Zhejiang University, Hangzhou 310058,
China

\textbf{Corresponding Author} : Address: Wuyi Wan, Zhejiang University,
Hangzhou 310058, China

Email: wanwuyi@zju.edu.cn

\textbf{Abstract}

This paper aims to solve the problem of accurately estimating flow
duration curves (FDC) in catchments lacking diachronic flow data. Based
on 645 sets of observed data in the middle and lower reaches of the
Yangtze River (YZR), which include 22 basin characteristic variables,
eight machine learning (ML) models (SVM, RF, BPNN, ELM, XGB, RBF,
PSO-BP, GWO-BP) were integrated to predict the FDC (quantiles of flow
rate corresponding to 15 exceedance probabilities were studied), after
which the model most suitable for predicting was determined. Finally,
the SHapley Additive exPlanation (SHAP) method was used to determine and
quantify the impact of various input variables on different quantiles
and the degree of that influence. Results indicate that: (1) The GWO-BP
model is the best ML model for predicting FDC among the eight, having
good prediction performances throughout the entire duration with
determination coefficients (\emph{R} \textsuperscript{2}) on the testing
set of 0.86 to 0.94 and Nash-Sutcliff criterion (\emph{NSE} ) of 0.78 to
0.94. (2) The ML model (BPNN) optimized using swarm intelligence can
effectively predict FDC. (3) The predictive impact of variables on
different quantiles varies, with and BFI\_mean contributes significantly
to predicting FDC. The former has a negative effect on the prediction
result and has better contribution to predicting higher flow rate (i.e.,
having higher accuracy in predicting the upper tail of FDC), whereas the
latter is the opposite. SHAP's explanations are consistent with the
physical model, revealing local interactions between predictive factors.
The results demonstrate that the method proposed in this paper can
greatly improve the prediction accuracy and is innovative and valuable
in model interpretation and factor selection.

\textbf{Keywords:} Flow duration curve (FDC), Streamflow quantile, Basin
characteristics, Machine learning, SHAP

\textbf{INTRODUCTION}

\textbf{1.1Estimating flow duration curves in ungagged catchments}

The flow duration curve (FDC) is defined as the relationship between a
certain flow rate and the frequency greater than or corresponding to
that flow rate during certain periods of time. It is essentially a
cumulative distribution function (Searcy, 1959), which comprehensively
describes the entire characteristics of runoff in a basin from low flow
to flood, and can better reflect the precipitation and runoff conditions
of the basin (Cheng et al., 2012). However, many water resource projects
are often located in areas without measured runoff data which leads to
difficulty to directly obtain the FDC (Veber Costa, 2020). Regional
analysis methods can be used to obtain the regional FDC from the areas
with measured runoff data, and convert it to areas without measured
runoff data to meet the design needs of water resource projects in that
area, which is of great help for water resource planning, design, and
runoff prediction in areas without measured data (Manuel Almeida, 2021;
Veber Costa, 2020; Mancini, 2016; Li et al., 2010; Croker K M, 2003).

There are usually two existing methods to establish FDC in unmeasured
areas: process and statistical based method (Bl\selectlanguage{ngerman}öschl, 2013) Based on
process-based method, the probability distribution of daily flow is
simulated by establishing a hydrological model of the watershed,
considering the characteristics of the watershed and the physical
mechanisms of the hydrological process (Cheng et al., 2012; Yokoo and
Sivapalan, 2011; Ceola et al., 2010; Botter et al., 2007; Doulatyari et
al., 2015) Statistical based method, on the other hand, are modeling
methods based on statistical models and data analysis. It infers future
hydrological variables by analyzing the statistical characteristics and
patterns of historical observed data, which does not require in-depth
understanding of the physical mechanisms of hydrological processes and
typically require a large amount of data for training and optimization
(Burgan and Aksoy, 2022a; Müller et al., 2014; Atieh et al., 2017a). The
advantage of process-based models is that they can analytically derive
the probability density function of flow and independently simulate the
impact of climate or geomorphic changes on FDC, with reliability under
non-steady conditions (Ghotbi et al., 2020; Ghotbi et al., 2020) . Its
disadvantage is that the assumption of spatial homogeneity in the
watershed makes its applicability relatively low (Leong and Yokoo, 2021)
. However, parameter estimation for process-based models is less
demanding, and can be determined using information such as rainfall,
climate, and geomorphic characteristics of the watershed at any location
with data (Schaefli et al., 2013; Karst et al., 2019). Based on existing
analysis and research on process-driven method, the physical
characteristics of the basin (such as average temperature, potential
evapotranspiration, elevation, etc.) distribute precipitation to various
parts of the river: groundwater recharge and base flow, surface runoff,
and rainstorm flow (Rice and Emanuel, 2017; Ye et al., 2012) .
Therefore, the precipitation accumulated and the features of the basin
will mainly influence the shape of FDC (Luan et al., 2021).

However, due to the uncertainty in runoff and climate mechanisms, this
method has limited application in areas without data (Reichl and Hack,
2017) . In addition, statistical methods often have better estimation
performance on FDC than process-based methods (Engeland and Hisdal,
2009; Over et al., 2018) . It mainly include (1) Using regression
methods to independently estimate quantiles through basin
characteristics (Farmer and Vogel, 2016) (2) Estimating statistical
moments and fitting the FDCs using appropriate distribution functions,
finding the relationship between the statistical parameters of the
function and the basin features (Almeida et al., 2021; Burgan and Aksoy,
2022b; Shin and Park, 2023) ; (3) Using streamflow index-based method
(Atieh et al., 2017b). (4) Using geostatistical method ,etc (Goodarzi
and Vazirian, 2023) . By comparing these two methods, it has been found
that the statistical approach is more sensitive to spatially sparse
data, while the process-based approach is more sensitive to observations
that are temporally limited (Müller and Thompson, 2016). Although
statistical methods often provide better FDC predictions than
process-based methods, it is obvious that they typically need a
significant amount of post-processing to explain the physical
untrustworthiness in the results. There have been numerous studies
attempting to combine process-based models and data-driven models in
hopes of fully leveraging their respective advantages to improve
prediction accuracy. For example, the relationship between quantiles and
watershed features was studied to ensure the monotonicity of quantile
estimation and explore the relationship between quantiles and their
related basin features (Requena et al., 2018; Poncelet et al., 2017) .

\textbf{1.2 Artificial intelligence methods in the prediction of FDCs}

With the development and maturity of data science and artificial
intelligence, the research focus of hydrological prediction models has
gradually shifted from process-drive to data-driven models
(Mohammadrezapour et al., 2019; Sharifi Garmdareh et al., 2018) . The
data-driven model was based on the statistical properties of the data,
without considering the physical causes of runoff, and directly
calculates the correlation between the input and output of the model to
obtain hydrological prediction results. Machine learning models
typically exhibit a relatively complex model structure. By adjusting
parameters and conducting model training, the model can continuously
approach the optimal mapping relationship between the input and output,
and the predicted results usually have high accuracy. However, due to
the limitations of the ``black box'', decision-makers cannot directly
know how machine learning models calculate decision results (Cortez and
Embrechts, 2013) . The ''black box'' of machine learning models
simplifies model input and training, which makes its prediction results
lack practical physical significance, and the model is unable to explain
how to obtain prediction results from the causes and mechanisms of
runoff formation, resulting in low credibility in practical prediction
work. But machine learning methods are widely used in hydrology (Khan et
al., 2016; Khan et al., 2019) because they have unreasonable
effectiveness when applied to real-world problems (Shen, 2018) . Due to
the complexity of hydrological systems which cannot be easily
represented by simple conceptual relationships between variables and the
nonlinear relationship between watershed characteristics and
hydrological characteristics, traditional methods lack sufficient
ability to predict FDCs, while artificial intelligence models have some
applicative potential (Nearing and Gupta, 2015) .

SVM, ANN, and nonlinear regression (NLR) were used for regression
prediction using different runoff duration as output variables and six
basin feature selections as input variables in a study of 33 watersheds.
The results indicate that SVR is the most suitable model for estimating
FDC (Vafakhah and Khosrobeigi Bozchaloei, 2020) . A multi-output neural
network model was developed to predict the FDC of 9203 dataless areas in
the southeastern United States over a 60-year period from 1950 to 2009,
suggesting that compared with single-output neural-network models,
multi-output neural networks is capable of learning monotonic
relationships between adjacent quantiles and yield better predictions
(Worland et al., 2019) .

Machine learning (ML) has demonstrated outstanding performance in
forecasting FDC and is extensively utilized for predicting (Ley et al.,
2023; Vaheddoost et al., 2023) . Existing research has primarily
concentrated on enhancing the prediction accuracy of FDC through single
ML model, neglecting the impact of its influencing factors, and the
prediction accuracy through traditional prediction methods is relatively
low. Moreover, there are few research of using multiple machine model
algorithms for comprehensive comparison, and conducting regionalization
research on FDC prediction based on geographical and climatic
characteristics. Explainable machine learning (eg. SHAP) is a rapidly
developing subfield aimed at understanding how models use inputs for
prediction and eliminating the black box problem (Kim, 2017) . Thus, the
main issues studied in this paper include (see \textbf{Figure 1} ):

{[}Insert Figure 1{]}

\textbf{Figure 1 Framework of the prediction and inference of FDC using
ML}

This paper utilizes a total of 645 sets of samples, made up of 22 basin
characteristic variables (including ``mutable'' and ``immutable'') in 30
years from 244 hydrometric stations located in the middle and lower
reaches of the Yangtze River basin. Using typical characteristics of the
basin, regional FDC model was established through machine learning
methods and the performance of these methods was compared to determine
the most suitable model for predicting the FDC. Firstly, the model
includes 22 basin characteristics that were selected and divided into
mutable and immutable variables and 15 corresponding quantiles of FDC.
Secondly, basin characteristic variable-flow quantile database was
established using eight typical ML models to study the nonlinear
relationship between the input parameter (basin characteristics) and the
fifteen flow quantiles which affect the shape of the FDC. Each quantile
was predicted and Taylor plots were applied to compare different ML
models to select the best one to estimate FDC. Finally, the key
influencing factors of various input on the fifteen quantiles were
quantified and determined using SHAP. How these important hydrological
factors affect the results was also discussed.

\textbf{2 DATA AND METHODS}

\textbf{2.1 Study area}

The Yangtze River Basin (YRB) has a well-developed water system, with
four main tributaries: the Yalong River, the Minjiang River, the Jialing
River, and the Han River. This study was conducted in the middle and
lower reaches of the YRB with the area of
1.8*10\textsuperscript{6}km\textsuperscript{2}, located from 90°33' to
122°25' E longitude and 24°30' to 35°45'N latitude (see \textbf{Figure
2} ) (Li et al., 2021).

{[}Insert Figure 2{]}

\textbf{Figure 2 Study area and location of 224 hydrometric stations}

\subsection*{2.2 Data}

{\label{data}}

The daily streamflow data of 267 hydrometric stations was downloaded
from the Annual Hydrological Report of the People's Republic of China,
in which 224 hydrometric stations with 30 years records from 1970 to
1990 and from 2007 to 2016 were selected (see \textbf{Figure 3} ).
According to the China Meteorological Data Network
(https://data.cma.cn/), daily precipitation, potential evaporation and
temperature data from 1961 to 2016 were downloaded from 698 evenly
distributed weather stations. The observed precipitation, potential
evaporation and temperature data were interpolated into the whole
Yangtze River basin with the method of Thiessen polygon (Meena et al.,
2013). The interpolated precipitation, potential evaporation and
temperature data of the basin area corresponding to 224 hydrometric
stations were averaged. From the geospatial data cloud
(\emph{http://www.gscloud.cn/}), the 30-meter digital elevation model
(DEM) was downloaded. Data on 361 reservoirs located in the mid-lower
reaches of the YZR were retrieved from the Global Reservoir and Dam
database (GRanD) (Lehner et al., 2011) .

{[}Insert Figure 3{]}

\textbf{Figure 3 Streamflow record for 224 hydrologic stations. We here
used complete decadal streamflow data for year 1970--1990 and
2007-2016.}

\textbf{2.3 FDC and its corresponding streamflow percentiles}

In order to present the characteristics of interdecadal changes, the
changes of every 10 years: 1970-1979, 1980-1989, 2007-2016 were counted
for total 30 years, providing more observational data for
regionalization. Runoff data includes 15 percentiles corresponding to
exceedance probabilities:\emph{Q} \textsubscript{0.3}, \emph{Q}
\textsubscript{0.5},\emph{Q} \textsubscript{1}, \emph{Q}
\textsubscript{5},\emph{Q} \textsubscript{10}, \emph{Q}
\textsubscript{20},\emph{Q} \textsubscript{30}, \emph{Q}
\textsubscript{50},\emph{Q} \textsubscript{70}, \emph{Q}
\textsubscript{80},\emph{Q} \textsubscript{90}, \emph{Q}
\textsubscript{95},\emph{Q} \textsubscript{99}, \emph{Q}
\textsubscript{99.5},\emph{Q} \textsubscript{99.7}, representing 0.3\%,
0.5\%, 1\%, 5\%, 10\%, 20\%, 50\%, 70\%, 80\%, 90\%, 95\%, 99\%, 99.5\%,
99.7\% respectively. These 15 quantile flow values are calculated and
are directly related to the return period. Each quantile represents a
different part of FDC, ranging from particularly high flow (\emph{Q}
\textsubscript{0.3},\emph{Q} \textsubscript{0.5}, \emph{Q}
\textsubscript{1} and\emph{Q} \textsubscript{5}) to particularly low
flow (\emph{Q} \textsubscript{95}, \emph{Q} \textsubscript{99},\emph{Q}
\textsubscript{99.5} and \emph{Q} \textsubscript{99.7}). For example,
taking Wuxi station as an example (see \textbf{Figure 4} ) and draw its
fitting curve using gamma distribution, the quantile corresponding to
Q\textsubscript{5} is 35.1160m\textsuperscript{3}/s, and the return
period of 35.1160m\textsuperscript{3}/s is a 20-year return period
(1/p=1/0.05=20). It is noteworthy that each quantile was log transformed
(log\textsubscript{10}(streamflow)) prior to modelling.

The variation characteristic of the \textbf{Figure 4} shows that
although the quantiles are randomly selected, they are used to determine
the overall shape of each FDC. Because low and high flow rates are
usually more important for drought and flood research, which are related
to many processes occurring in ecosystems, the selected exceedance
probabilities are closer to both ends to explain very large or very
small changes of corresponding quantiles, while the distribution of the
center part of the curve is determined by exceedance probabilities which
are distributed more evenly and fewer. The overall variation of the
curve is significant, with both the high and low flow parts of the curve
showing a downward trend, and the FDC showing an S-shape. The change at
the low tail of the curve is greater than the change at the front of the
curve, indicating that the low flow part has a greater change.

{[}Insert Figure 4{]}

\textbf{Figure 4 Streamflow values corresponding to different
probability of exceedance (Year 1980-1989\_Wuxi Station). Red marks
represent the point of 15 streamflow percentiles analyzed in this
paper.}

For a more intuitive presentation, the final dataset includes a 10-year
combination of 224 sites, with each quantile normalized by the
logarithmic coordinate transformation (log (quantile)). The 80\% of
randomly selected stations were used as training sets, while the
remaining 20\% were reserved for testing.

\textbf{2.4 Machine learning algorithms}

2.4.1 Support vector machine (SVM)

SVM has developed from the optimal classification surface in linearly
separable cases, and based on statistical learning theory, it has
excellent generalization ability in machine learning by replacing the
empirical risk minimization principle with the structural risk
minimization principle (Araghinejad, 2013) .By introducing appropriate
inner product kernel functions, the samples in the input space can be
mapped to high-dimensional spaces, thereby achieving linear
classification or regression after a certain nonlinear transformation
without increasing computational complexity (see \textbf{Figure 5} (a)).
When SVM is applied to regression problems, its learning goal is to find
the best hyperplane closest to all data points at a given interval.
Given the dataset \{(x\textsubscript{1}. y\textsubscript{1}),
(x\textsubscript{2}. y\textsubscript{2}), \ldots{}, (x\textsubscript{i},
y\textsubscript{i})\}, the problem can be converted into an optimization
problem under given objective functions and constraint which shows as
followed:

Where, and b are the weight coefficients and bias coefficients of the
optimal hyperplane respectively. C is the penalty factor, \begin{CJK}{UTF8}{gbsn}、 \end{CJK}\selectlanguage{english}are
introduced slack variables; is the given interval (ARNARI S, 1999;
Choubin et al., 2018; VAPNIK V, 1997) .

2.4.2 Random forest (RF)

Breiman improved the regression tree model based on the Bagging
algorithm and proposed the random forest algorithm (BREIMAN, 2001) ,
which consists of sub-training sets and sub-regression models (decision
trees), which extracts \emph{m} multiple sample data points from the
original sample set \emph{D} through Bootstrap resampling method to form
a sub-training sample set with the same sample size as the original one
(see \textbf{Figure 5} (b)). For each sub-training sample set, a
sub-regression model is constructed, which is called random forest model
(Das et al., 2017; Ibarra-Berastegi et al., 2015; Nashwan and Shahid,
2019) .

2.4.3 Back propagation neural network (BPNN)

BPNN is a kind of multi-layer feedforward neural network used for
nonlinear functions, which trains the weights and thresholds for many
times. When the actual output does not match the observed, it enters the
error backpropagation stage: it updates the weights between the hidden
layer and the input layer, as well as the thresholds of the hidden layer
based on the error of the output values (Maier and Dandy, 2000) . In
this paper, the sigmoid (\emph{x} ) function serves as the activation
function for input-hidden layer and hidden-output layer (see
\textbf{Figure 5} (c)), and the formula is as followed:

2.4.4 Extreme learning machine (ELM)

ELM is a feedforward neural network with only single hidden layer (HUANG
et al., 2004) , which differs from the BP algorithm in that it does not
need to adjust the weight of the hidden layer through reverse iteration.
The weight of the input feature vector is randomly assigned from the
input layer to the hidden laye. (Huang et al., 2006) (see \textbf{Figure
5} (d)). Its difference with BP is that the weights between the layers
all need to be iteratively solved using the gradient descent method in
the BP algorithm, But for ELM, the weight of input layer and the hidden
layer can be determined without iteration for the certain input feature
vector.

\subsubsection*{\texorpdfstring{\emph{2.4.5 Extreme gradient boosting
(XGB)}}{2.4.5 Extreme gradient boosting (XGB)}}

{\label{extreme-gradient-boosting-xgb}}

The Extreme gradient boosting (XGB) algorithm reduces the error of the
previous prediction step by continuously generating new regression
trees, gradually narrowing the gap between the true and predicted
values, and thereby improving the prediction accuracy (Chen and
Guestrin, 2016) (see \textbf{Figure 5} (e)). It improves the
generalization ability and computational efficiency of the model by
introducing regularization terms and parallel computing techniques on
the basis of the original gradient boosting decision tree (GBDT)
algorithm.

\subsubsection*{\texorpdfstring{\emph{2.4.6 Radial-Basis Function
(RBF)}}{2.4.6 Radial-Basis Function (RBF)}}

{\label{radial-basis-function-rbf}}

Radial-Basis Function (RBF) network can approximate any nonlinear
function, deal with the difficulty to analyze regularity in the system
(Majnooni et al., 2023) (see\textbf{Figure 5} (f)). Compared with BP,
RBF has only one hidden layer, while BP does not limit the number of it.
BP is a global approximation of nonlinear mapping, while RBF is a local
approximation of nonlinear mapping, with faster training speed.

{[}Insert Figure 5{]}

\textbf{Figure 5Principles of the six machine learning algorithms}

\subsubsection*{2.5 Swarm intelligence optimization
algorithm}

{\label{swarm-intelligence-optimization-algorithm}}

Although BP neural networks can quickly adapt to various problems, they
are also prone to falling into local optimum and overfitting, which will
affect the prediction results, while swarm intelligence optimization
algorithms demonstrates their effectiveness in solving complex
optimization problems (Cai et al., 2018). Therefore, we present swarm
intelligence optimization algorithms to seek out both optimal global and
local solution. These algorithms generate initial populations and
utilize heuristic rules to carry out searches and reduce overfitting
risks by optimizing weight parameters to improve the accuracy of FDC
prediction.

\subsubsection*{\texorpdfstring{\emph{2.5.1 Particle swarm
optimization}and \emph{gray wolf optimizer
algorithm}}{2.5.1 Particle swarm optimizationand gray wolf optimizer algorithm}}

{\label{particle-swarm-optimization-and-gray-wolf-optimizer-algorithm}}

The particle swarm optimization algorithm (PSO) originates from a group
of animal social interaction models that search for food (Poli et al.,
2007) , in which birds are regarded as particles, and all particle
information within the group is shared to find the optimal strategy
(see\textbf{Figure 6} (a)). Assuming a fixed search area is limited to
a\emph{d} -dimensional space, where the number of particles is \emph{n}
. During the iteration process, particles track the ``individual
extremum '' and ``global extremum '' by changing their own speed and
position. The formula for particle speed and position is as follows:

Where and represent the velocities of particle at the iteration and ,
having both size and direction; Similarly, and are the positions of
those particles; is the individual extremum and is the global extremum
which is the uniformly distributed random number of the currently found
optimal solution in the particle swarm. and denote acceleration
coefficients; and are two random numbers between 0 and 1; is the inertia
weight.

The Grey Wolf Optimizer (GWO) boasts several advantages, including
simplicity in structure and parameters, and capabilities in adaptive
adjustment. These features render it an effective tool in striking a
balance between local and global optimization (Dehghani et al., 2019;
Seifi and Soroush, 2020) . It includes three main stages: finding,
surrounding, and attacking preys (Mirjalili et al., 2014)
.\textbf{Figure 6} (b) shows the working structure. \begin{CJK}{UTF8}{gbsn}、、 \end{CJK}\selectlanguage{english}and the
remaining wolves dominate the social hierarchy, and the level of
dominance of wolf species descends from to , which means that is the
most powerful wolf category, with wolves \begin{CJK}{UTF8}{gbsn}、 \end{CJK}\selectlanguage{english}and guiding the remaining
wolves to search towards their targets. GWO can bypass local optimal
stagnation and enhance the global optimal convergence ability (Adnan et
al., 2023) (Adnan et al., 2023). The hunting behavior of gray wolves
encircling their prey is quantified using the following position update
formula (Maroufpoor et al., 2020; Zhou et al., 2021) .

Where and represent the positions of the prey and the grey wolf
respectively; t represents the current number of iteration; D represents
the distance between the wolf and its prey; is the wolf's position at
time (\emph{t} +1); C and A are coefficient vectors; The above equation
represents the distance between the grey wolf and its prey, and the
following formula represents the the grey wolves' position updated.

{[}Insert Figure 6{]}

\textbf{Figure 6 The principles of swarm intelligence optimization
algorithm: PSO and GWO}

\subsubsection*{2.6 Evaluation of model
performance}

{\label{evaluation-of-model-performance}}

The performance evaluation criteria including \emph{R} -squared
(\emph{R} \textsuperscript{2}), root mean squared error (RMSE), and
Nash-Sutcliffe Efficiency (NSE) (Nash and Sutcliffe, 1970) were used to
evaluate the ML performance.

where and are the predicted and the observed value; and are the average
values of observed and predicted; \emph{n} represents the sample number.

\subsubsection*{2.7 Explanation of ML's
output}

{\label{explanation-of-mls-output}}

SHapley Additive exPlanation (SHAP) is based on the Shapley value in
game theory, which can take into account the mutual influence between
all features, providing more accurate and comprehensive explanations.
For each prediction sample, the model generates a corresponding
prediction value, and the SHAP value represents the contribution of each
feature to the model's overall prediction (Lee, 2017) . The Shapley
value was interpreted as an additive feature attribution method.

\subsubsection*{3 DATABASE DESCRIPTION AND
ANALYSI}

{\label{database-description-and-analysi}}

\subsubsection*{3.1 Selection of input
variables}

{\label{selection-of-input-variables}}

Though multiple research have been carried out on the prediction of flow
rate in the regions without historical flow rate data via FDC, the
accuracy of prediction and applicability to regions is still unsettled.
Therefore, 224*3 sets of test data were used in this paper. All basin
characteristics selected in this paper (Table 1) which were selected as
input variables are divided into two categories: mutable variables and
immutable variables. Characteristics that may change over time (such as
Aridity index (\emph{AI} ), precipitation, etc.) are regarded to as
``mutable'' variables. Those characteristics which are reflected as
quantities that are fixed over time, for example, such as
elevation\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}drainage area\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}reservoir control area\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}average slope
(dimensionless)\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}baseflow index (dimensionless)\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}topographic wetness
index (dimensionless) et al., are regarded as ``immutable'' variables.
The impact of basin characteristics on FDC seems to be empirical (Elena
Ridolf et al., 2020). By assessing the value of each parameter, 15
unique corresponding flow percentile values can be obtained.

\textbf{Table 1 Basin characteristic parameters selected for basins of
hydrometric stations}

{[}Insert Table 1{]}

\subsubsection*{3.2 Correlation and description of basin
characteristics}

{\label{correlation-and-description-of-basin-characteristics}}

The results judged by the Spearman correlation coefficients
(see\textbf{Figure 7} ) show that BFI\_ Mean and has a significant
correlation with flow data includes 15 percentiles corresponding to
exceedance probabilities. Smax, AP\_max, DP\_ std, ATP and SWS all
exhibit a negative correlation with the streamflow percentile. From
\textbf{Figure 7} (c), the relationship between DA and the streamflow
percentile varies from negative to positive with the rise of percentiles
(decrease in the rate of flow and the return period), while the
relationship between wind and flow percentile is reversed. Moreover,
BFI\_ Mean has a better correlation with low flow rate (bottom part of
the FDC curve), while has a higher correlation with high flow rate.

{[}Insert Figure 7{]}

\textbf{Figure 7 Correlation of input and output variables. (a) shows
the correlation between variables of selected characteristics and Q0.3.
(b) shows the correlation between all the variables. (c) shows the
correlation between variables of selected characteristics and the flow
rate includes 15 streamflow percentiles corresponding to exceedance
probabilities.}

{[}Insert Figure 8{]}

\textbf{Figure 8 Distribution of the seven variables with high
correlation with\emph{Q}\textsubscript{0.3}}

As shown in \textbf{Figure 8} , scatter matrix are further used to
display the data distribution features of these input variables.
Considering the large size of the matrix, here only a small portion of
the matrix was shown, selectively analyzing the seven variables
(BFI\_mean\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}Smax\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}ATP\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}DP\_std\begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}AP\_max\begin{CJK}{UTF8}{gbsn}、、\end{CJK}\selectlanguage{english}SWS) with high correlation
with \emph{Q} \textsubscript{0.3} in \textbf{Figure 7} to show the data
situation. The frequency distribution of these input variables are
represented in the diagonal line. The upper right section portrays the
scatter plot of input basin characteristics. It can be observed that ATP
and DP\_std follow a linear relationship. Nevertheless, the variables
exhibit a tendency towards nonlinear distribution, thus uncomplicated
linear regression and nonlinear regression models are not able to
precisely describe the distribution features of the input variables and
their impacts on the output variables. The lower left section displays
the probability density of the distribution of these input variables,
with darker colors indicating a higher probability of data appearing. To
take an instance, ATP is concentrated near the value of 500 when
BFI\_mean assumes the value of 0.5. In order to fully evaluate their
impacts on them, SHAP was utilized to increase the interpretability of
ML models and disclose the mechanism in section 4.4.

\subsubsection*{4 RESULTS AND DISCUSSIONS}

{\label{results-and-discussions}}

\subsubsection*{4.1 Parametric analysis}

{\label{parametric-analysis}}

{[}Insert Figure 9{]}

\textbf{Figure 9}

\textbf{Parametric optimization of the algorithm (SVM and RF)}

ML models' performance is influenced by their parameters, making it
essential to optimize the parameters by pre-setting the value of MSE.
There are several methods adopted to address the poor prediction
accuracy and acquire the optimal parameter.

(1) There are two ways to adjust the network parameters of the BPNN: (a)
Determining the minimum MSE value by comparing different numbers of
hidden layers or nodes in the hidden layer to obtain the optimal
parameters. (b) Algorithms of PSO and GWO were combined for parameter
optimization and the optimization ability of two methods was compared
in\textbf{Figure 17} .

(2) The penalty coefficient \emph{c} and gamma \emph{g} play an
important role in the SVM model, where \emph{c} determines the
generalization ability and \emph{g} affects the prediction accuracy. The
libSVM toolbox (Chang and Lin, 2001) is adopted for parameter
optimization of \emph{c} and \emph{g} , the process of which is shown
in\textbf{Figure 9} (a).

(3) In RF model, leaves' number has an impact on the prediction accuracy
and grown trees determine whether the model will be over-fitted. The
process of optimization are shown in \textbf{Figure 9} (b).

(4) The optimal number of nodes of the hidden layer is needed to be
determined in the ELM. Therefore, the models' optimization search is
completed by pre-setting MSE.

(5) The XGB optimizes parameters through regularization,
cross-validation, and so on.

\subsubsection*{4.2 ML predictive performance
analysis}

{\label{ml-predictive-performance-analysis}}

{[}Insert Figure 10{]}

\textbf{Figure 10 Comparison of predicted and observed
Q\textsubscript{0.3}}

{[}Insert Figure 11{]}

\textbf{Figure 11 Comparison of predicted and observed
Q\textsubscript{5}}

{[}Insert Figure 12{]}

\textbf{Figure 12 Comparison of predicted and observed
Q\textsubscript{50}}

{[}Insert Figure 13{]}

\textbf{Figure 13 Comparison of predicted and observed
Q\textsubscript{90}}

{[}Insert Figure 14{]}

\textbf{Figure 14 Comparison of predicted and observed
Q\textsubscript{99.7}}

{[}Insert Figure 15{]}

\textbf{Figure 15 Performance of 8 ML models on the training sets}

{[}Insert Figure 16{]}

\textbf{Figure 16 Performance of 8 ML models on the testing sets}

In this paper, a comparative analysis is conducted on the predicted and
observed values of 15 streamflow percentiles corresponding to the FDCs
obtained from 8 models, and the predicted and observed values show
consistency. From \textbf{Figure 10} to \textbf{Figure 14} , we mainly
analyze the prediction results of five key streamflow percentiles (,
,,,).

Overall, the ratio of predicted to observed values is stable around 1,
and \emph{R} \textsuperscript{2} is close to 1. Neural networks have
better generalization capabilities than other machine learning
algorithms, as evidenced by their better predictive accuracy on the
testing set. The predictive accuracy of each model for the upper tail of
the FDC is higher than that for the lower tail, with\emph{Q}
\textsubscript{5} having the highest predictive accuracy and\emph{Q}
\textsubscript{99.7} having the lowest predictive accuracy. The
prediction difference between observed and predicted values may be
attributed to the random-like property of hydrological phenomena.
Related literature (Montanari and Koutsoyiannis, 2012) also reached
similar conclusions. Among the eight ML models on the testing set, the
prediction performance of ELM and RBF is worse, probably due to the
simplicity of single-layer neural networks. The predictive ability of
the XGB, PSO-BP and GWO-BP models is significantly better. We noticed
that these three models show good predictive ability on both the
training and testing set, with the \emph{R} \textsuperscript{2} for both
the training and testing set being greater than 0.8 at different
streamflow percentiles. The predictive performance of models shouldn't
only be evaluated by a single metric. Compared to scatter plots, which
can only display the relationship between individual indicators (Choubin
et al., 2018) , the Taylor diagram integrates three evaluation metrics:
correlation coefficient, centered root-mean-square, and standard
deviation, based on the cosine relationship between the three to
evaluate the predicting performance from different perspectives
(\textbf{Figure 15} \begin{CJK}{UTF8}{gbsn}、\end{CJK}\selectlanguage{english}\textbf{Figure 16} ). When the model prediction
results are consistent with the observed values, the closer the point
``model'' is to the point ``observed'' on the x-axis, the higher the
correlation between such models and observations.

{[}Insert Figure 17{]}

\textbf{Figure 17 Comparison of predicted value obtained by PSO-BP,
GWP-BP and observed data (Testing sets)}

For training sets, the RBF and ELM models' prediction performances are
poor, while XGB performs the best. For high tails, the prediction
performance of the eight models can be ranked as XGB \textgreater{} SVM
\textgreater{}RF \textgreater{} PSO-BP\textgreater{} BPNN \textgreater{}
GWO-BP \textgreater{} RBF \textgreater{} ELM, while the prediction
performance can be ranked as XGB \textgreater{} RF \textgreater{} BPNN
\textgreater{} GWO-BP \textgreater{} SVM \textgreater{}PSO-BP
\textgreater{} RBF \textgreater{} ELM for low-tailed data.

As for testing sets, the prediction performance of the BPNN is not as
good as the other seven models, while XGB, PSO-BP, and GWO-BP all
exhibit good performance. For high tails, the prediction performance can
be ranked as follows: PSO-BP \textgreater{} GWO-BP \textgreater{} XGB
\textgreater{} RF \textgreater{} SVM \textgreater{} BPNN \textgreater{}
ELM \textgreater{} RBF. For low-tailed data, the prediction performance
is ranked as follows: GWO-BP \textgreater{} XGB \textgreater{} PSO-BP
\textgreater{} RF \textgreater{}BPNN \textgreater{} ELM
\textgreater{}SVM \textgreater{}RBF.

It is worth noting that the prediction accuracy of the lower tail of FDC
through machine learning is significantly lower than that of the upper
tail, but GWO-BP and XGB perform well in predicting the lower tail. By
comparing evaluation indicators, it is determined that the GWO-BP and
XGB models are the best models for predicting FDC. Moreover, it can be
concluded that optimizing ML model parameters using the swarm
intelligence optimization algorithms can effectively and significantly
enhance the model's predictive capability and generalization ability by
comparing BPNN, PSO-BP, and GWO-BP (\textbf{Figure 17} ).

\subsubsection*{4.3 Prediction results throughout the entire
duration}

{\label{prediction-results-throughout-the-entire-duration}}

{[}Insert Figure 18{]}

\textbf{Figure 18 Overall evaluation (\emph{R}\textsuperscript{2})
(testing sets) of the estimated quality of ML models (15 streamflow
percentiles)}

Multiple points of streamflow percentiles can reflect the shape of the
FDC. The \emph{R} \textsuperscript{2} and NSE are usually used to assess
the model prediction. We believe that an \emph{R}
\textsuperscript{2}greater than 0.85 indicates good predictive
performance for the model. In addition to considering \emph{R}
\textsuperscript{2}, \emph{std} ,\emph{cor} , and \emph{R} MSE which we
have analyzed and discussed in the condition of six most typical
streamflow percentiles in the previous section, models with \emph{NSE}
values less than or equal to 0.50, 0.50\textasciitilde{}0.65,
0.65\textasciitilde{}0.75, and greater than 0.75 are considered to
represent 4 categories: bad, satisfactory, good, and excellent
performance respectively (Fatehi et al., 2015) .

{[}Insert Figure 19{]}

\textbf{Figure 19 Overall evaluation (NSE) (testing set) of the
estimated quality of ML models (15 streamflow percentiles)}

As shown in \textbf{Figure 18} and \textbf{Figure 19} , considering
the\emph{R} \textsuperscript{2} and \emph{NSE} criteria, the results
show that RF, PSO-BP, and XGB all achieve very good performance, except
for Q99.7 which only has satisfactory results. And it is observed that
the XGB model has less predictive power for larger and smaller
streamflow percentiles than for the middle streamflow percentiles, with
particularly good predictive performance for the middle part. The GWO-BP
model performs well across the entire duration range in the testing set
(i.e., high flow to low flow) with \emph{R} \textsuperscript{2} of 0.86
to 0.94 and \emph{NSE} of 0.78 to 0.94. The performance of the RF,
PSO-BP, and XGB models is also good throughout the entire duration, but
it is lower than that of the GWO-BP model. From the perspective of the
sustained range of the entire FDC, the GWO-BP is the best model to
predict FDC among all the models in this paper.

Compared with the research of Vafakhah and Khosrobeigi Bozchaloei
(Vafakhah and Khosrobeigi Bozchaloei, 2020) , which is believed that SVR
is the optimal model for predicting FDC with relative RMSE of 9.37 to
1.45 and \emph{NSE} of 0.54 to 0.91, the GWO-BP model we selected in
this paper greatly improve the accuracy of prediction.

\subsubsection*{4.4 Feature importance analysis of the
processes}

{\label{feature-importance-analysis-of-the-processes}}

{[}Insert Figure 20{]}

\textbf{Figure 20 Interpretation of the predicted FDC and feature
important analysis}

Due to the ``black box'' issue, the ML models have their limitations.
(Esterhuizen et al., 2022) . The ``feature importance'' merely reflects
which feature is more important, but how it influences the prediction
results is unknown. In this paper, Shapley was used to explain the
results of machine learning. The advantage of SHAP values is that they
not only reveal the impact of each feature in given samples but also
indicate the sign of that impact (i.e., whether it is positive or
negative) (Dikshit and Pradhan, 2021) .

From \textbf{Figure 20} , it can be seen that the impact of 22 variables
of basin characteristics on 6 critical streamflow percentiles (,,,,,)
was analyzed. The SHAP values are calculated for each sample and
variable, globally demonstrating the impact of feature on the model,
which quantifies the contribution of 22 environmental variables to
different streamflow percentiles. Each row represents an environmental
input variable, with the horizontal axis indicating the distribution of
SHAP values. Each point represents a sample, with color indicating the
feature value number (red for high values and blue for low values).

Comparing \textbf{Figure 20} (a) and Section 3.2, the correlation
coefficients between BFI\_mean, Smax, ATP, DP\_std, AP\_max, , SWS and
are relatively high, which are slightly different from the neural
network prediction results but generally consistent. It is worth paying
attention to that the zero value represents the average value of on the
horizontal axis. Considering , there is a rise in the value of SHAP when
decreases (changes in the color from red to blue). When reaches its
maximum, the is 1 lower than its average value, while reaches its
minimum, the is 2 higher than its average value. This is because a high
probability of no precipitation days indicates a decrease in
precipitation frequency (Cheng et al., 2012) , which will significantly
lower the value of flow rate quantile.

It also can be seen that the impact of environmental variables on
different streamflow quantiles varies noticeably. However, the two main
influencing factors for streamflow quantiles remain nearly unchanged,
with and BFI\_mean playing the key roles. High values of the will reduce
the flow rate of streamflow quantiles, exerting a negative impact, while
high values of the BFI\_mean feature will increase the flow rate of
streamflow quantiles, exerting a positive impact. with higher SHAP
values results in lower streamflow percentiles, exerting a negative
impact to the output values, while BFI\_mean with higher SHAP values
results in higher streamflow percentiles, having positive impacts on the
output values. Additionally, it can be observed that contributes more to
the prediction of high flow rate values such as ,, which means it will
predict the upper tail of FDC more accurately, while BFI\_mean has a
greater impact on the prediction of low flow rate values such as , which
means it will predict the lower tail of FDC more accurately.

The main influencing factors obtained through SHAP in this paper are
consistent with the physical controls of the gamma distribution fitting
parameters in the same region, proving the accuracy of the model in this
paper (Yu Zhou, 2023). The influence of annual average precipitation and
maximum precipitation on the prediction results of flow quantiles is not
significant, while the prediction results of flow quantiles are closely
related to , indicating that high flow may be driven by short-term
precipitation events, which are closely related to the frequency of
precipitation occurrence, and these events cannot captured by annual
average precipitation. The frequency of precipitation has a significant
impact on the prediction results of FDC, mainly because it directly
affects the runoff generation mechanism and water balance of the
watershed (Butcher et al., 2021). High precipitation frequency means
that there will be more precipitation in a shorter period of time, which
will lead to faster collection of surface runoff, the increase in
saturation degree of soil and reduction of the infiltration capacity of
soil (Crow et al., 2018). Soil moisture content is high and the
evaporation amount will decrease accordingly, resulting in reduced water
consumption. Thus, the negative impact of precipitation frequency on the
streamflow corresponding to percentiles may be mainly due to the fact
that frequent precipitation can lead to faster collection of surface
runoff, resulting in slower increases or faster decreases in the
streamflow. The BFI is largely influenced by the water storage capacity
of the aquifer and human activities. It indicates the importance of the
aquifer's water storage capacity in predicting low flow parts (Mazvimavi
et al., 2004) . Basins with low BFI cannot maintain good water flow
mobility. This may result in a shorter duration of flow in high flow
areas, while basins with high BFI can better maintain water flow
mobility, thus maintaining high flow conditions for a longer period of
time, which can explain why BFI\_mean exerts positive impacts to the
output values and has greater impacts on the prediction of low flow rate
values.

Like other statistical-based methods (Burgan and Aksoy, 2022c) , this
paper also has shortcomings of the subjectivity and uncertainty in
variables selection (Veber Costa, 2020) . In future research, except for
exploring more watershed characteristics that influence FDCs and
incorporating them into the model for more precise prediction, larger
datasets and scales (e.g., global scale) are needed to be considered and
examined to enhance the applicability of the model before it can be
applied to various watersheds with more diverse climate and landscape
conditions.

For most data-driven models, such as neural networks, only the
correlation between inputs and outputs is utilized, and the impact
mechanism of influencing factors is unknown (Atieh et al., 2017c;
Bozchaloei and Vafakhah, 2015) (Atieh et al., 2017c; Bozchaloei and
Vafakhah, 2015). Scholars pointed out that machine learning (ML) can
help hydrology make progress in many ways, including (1) incorporating
physics into ML models; and (2) improving the explanatory ability of ML
models (Shen, 2018) (Shen, 2018). From these two perspectives, the
findings of this paper can provide new methods and insights for more
accurately data-driven FDC curve prediction and analysis, which will
help provide scientific basis for water resource management and
hydrological forecasting and reveal the undelying physical processes.

\subsubsection*{5 CONCLUSION}

{\label{conclusion}}

This paper proposed the different ML methods to estimate FDCs. Based on
a total of 645 sets of samples, made up of 22 basin characteristic
variables (including ``mutable'' and ``immutable''), eight ML models are
integrated to predict the FDC (flow quantiles corresponding to 15
exceedance probabilities). Moreover, the SHAP analysis was used to
identify the main input variables that affect the prediction results of
different streamflow quantiles and the degree of that influence. The
optimal model for predicting under this environmental condition was
found. The main conclusions can be drawn in the following:

\begin{enumerate}
\tightlist
\item
  With the high prediction accuracy and good generalization ability,
  GWO-BP and XGB are the best models for predicting FDC. Moreover,
  optimizing ML model parameters using the swarm intelligence
  optimization algorithms can significantly enhance the model's
  predictive capability and generalization ability of the original BPNN.
\item
  From the perspective of the sustained range of the entire FDC, GWO-BP
  is the best predictive model among the eight with \emph{R}
  \textsuperscript{2} of 0.86 to 0.94 and\emph{NSE} of 0.78 to 0.94 in
  the testing set. It significantly improved the prediction accuracy of
  existing research, which is believed that SVR is the optimal model for
  predicting FDC with RMSE of 9.37 to 1.45 and \emph{NSE} of 0.54 to
  0.91 (Vafakhah and Khosrobeigi Bozchaloei, 2020) .
\item
  The predictive impact of variables on different quantiles varies, with
  and BFI\_mean contributes the most significantly to predicting FDC.
  The has negative effects on the prediction result and has better
  contribution to predicting higher flow rate, which is mainly due to
  the fact that frequent precipitation can lead to faster collection of
  surface runoff, resulting in slower increases or faster decreases in
  the streamflow. Basins with low BFI cannot maintain good water flow
  mobility, which may result in a shorter duration of flow in high flow
  areas, while basins with high BFI can better maintain water flow
  mobility, thus maintaining high flow conditions for a longer period of
  time. Therefore, BFI\_mean exerts positive impacts to the output
  values and has a greater impact on the prediction of low flow rate
  values.
\end{enumerate}

\textbf{ACKNOWLEDGEMENTS}

The authors are deeply grateful for the support provided by the National
Natural Science Foundation of China (No. 52379080 and No. 52079122)

\textbf{DATA AVAILABILITY}

Daily precipitation and temperature data (1961-2016) were from the China
Meteorological Data Network (https://data.cma.cn/). The 30-meter digital
elevation model (DEM) was from the geospatial data cloud
(http://www.gscloud.cn/). Data on 361 reservoirs located in the
mid-lower reaches of the YZR were retrieved from the Global Reservoir
and Dam database (GRanD) (Lehner et al., 2011) .

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